MTH3008 Weekly Problems 2

Original Documents: Problem Sheet / My Handwritten Solutions / Provided Solutions

Vibes: Lots of repetition and using the basic definitions in fun ways - pretty cool little practical.

Used Techniques:

• Definition/properties of Curl, Divergence, and Gradient.
• Converting/simplifying Suffix Notation and vector notation using the Kronecker Delta and Alternating Tensor.
• Using the fact that $r=(x_j{x}_j{)}^{1/2}$.
• Partial differentiation of suffix notation.

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2.1. Validity of Differential Operator Combinations

Question

Which of the following combinations of vector differential operators are valid?

1. Curl curl, curl grad, div grad.
2. Div grad, div curl, div div.
3. Grad div, curl grad, curl curl, grad grad, div curl.
4. Div grad, div curl, curl grad, div curl.

First and fourth, by using input/output properties of Curl (vector -> vector), Divergence (vector -> scalar), and Gradient (scalar -> gradient), as well as composition of functions working right -> left.

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2.2. Suffix Notation to Vector Notation

Question

Translate the suffix notation equation, ${\delta }_{ij}{c}_j+{ϵ}_{kji}{a}_k{b}_j=d_{\ell }{e}_m{c}_i{b}_{\ell }{c}_m$, into ordinary vector notation.

Using the properties of the Alternating Tensor and Kronecker Delta, as well as basic Suffix Notation, we can determine that $\boxed{\mathbf{c}+(\mathbf{a}+\mathbf{b})=\mathbf{c}(\mathbf{b}\cdot \mathbf{d})(\mathbf{c}\cdot \mathbf{e})}$.

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2.3. Simplify Suffix Notation Expressions

Question

Let $\mathbf{u}$, $\mathbf{v}$, $\mathbf{w}$, and $\mathbf{z}$ be vectors in ${\mathbb{R}}^3$. Using suffix notation, find an expression involving no cross products for $(\mathbf{u}\times (\mathbf{v}\times \mathbf{w})\cdot \mathbf{z})$.

Write your final answer in vector notation. Provide all steps of your workings.

Slowly going through the expression, converting to Suffix Notation, and then simplifying using the properties of the Alternating Tensor and then Kronecker Delta after relation, yields $\boxed{(\mathbf{u}\cdot \mathbf{w})(\mathbf{v}\cdot \mathbf{z})-(\mathbf{u}\cdot \mathbf{v})(\mathbf{w}\cdot \mathbf{z})}$.

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2.4. Find the Gradient of a Dot Product

Question

Compute the gradient of a dot product, i.e. $\nabla (\mathbf{u}\cdot \mathbf{v})$, that is…

1. Show that $[\mathbf{u}\times (\nabla \times \mathbf{v}){]}_i=u_j\frac{\partial v_j}{\partial x_j}-u_j\frac{\partial v_i}{\partial x_j}$.
2. Use item (1) to show that $[\mathbf{u}\times (\nabla \times \mathbf{v})+\mathbf{v}\times (\nabla \times \mathbf{u}){]}_i=u_j\frac{\partial v_j}{\partial x_j}-u_j\frac{\partial u_i}{\partial x_j}+v_j\frac{\partial u_j}{\partial x_j}-v_j\frac{\partial v_i}{\partial x_j}$.
3. Conclude that $\nabla (\mathbf{u}\cdot \mathbf{v})=\mathbf{u}\times (\nabla \times \mathbf{v})+\mathbf{v}\times (\nabla \times \mathbf{u})+(\mathbf{u}\cdot \nabla )\mathbf{v}+(\mathbf{v}\cdot \nabla )\mathbf{u}$.

Loads of effort, but basically convert to Suffix Notation then simplify with the Alternating Tensor then Kronecker Delta through their relation, as always. Then convert to using Gradient, Divergence, and Gradient, all within vector notation, and rearrange to give the final answer.

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2.5. Check Relations Involving Tensors

Question

Recall the relation ${ϵ}_{ijk}{ϵ}_{k\ell m}={\delta }_{i\ell }{\delta }_{jm}-{\delta }_{im}{\delta }_{j\ell }$ and check it for the following cases…

1. $i=1,j=2,k=3,\ell =1,m=2$,
2. $i=2,j=1,k=3,\ell =2,m=1$.

Substitute then use the definitions of Alternating Tensor for the left side, then Kronecker Delta for the right. Valid for both cases.

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2.6. Simplify Suffix Notation Expressions

Question

Simplify the following suffix notation expressions…

1. ${\delta }_{ij}{ϵ}_{ijk}$,
2. ${ϵ}_{ijk}{ϵ}_{i\ell m}$,
3. ${ϵ}_{ijk}{ϵ}_{ijm}$,
4. ${ϵ}_{ijk}{ϵ}_{ijk}$.

Note: these are all vectors.

1. Use Kronecker Delta to substitute which then creates a duplicated index in Alternating Tensor, hence gives $\boxed{0}$.
2. Use the relation between the Alternating Tensor (making sure the repeated indices are touching) and Kronecker Delta to give ${\delta }_{j\ell }{\delta }_{km}-{\delta }_{jm}{\delta }_{k\ell }$.
3. Use the relation between the Alternating Tensor (making sure the repeated indices are touching) and Kronecker Delta, then simplifying using each definition to get $\boxed{2{\delta }_{km}}$.
4. Use the relation between the Alternating Tensor (making sure the repeated indices are touching) and Kronecker Delta, then simplifying using each definition to get $\boxed{6}$.

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2.7. Prove Matrix Relations

Question

Use the formula, ${ϵ}_{pqr}|M|={ϵ}_{ijk}{M}_{pi}{M}_{qj}{M}_{rk}$, to show that…

1. $6|M|={ϵ}_{pqr}{ϵ}_{ijk}{M}_{pi}{M}_{qj}{M}_{rk}$,
2. $|M^T|=|M|$,
3. $|MN|=|M||N|$.

1. Use the given formula, rearrange, and simplify using the Alternating Tensor.
2. Relabel showing equivalency.
3. Same vibes.

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2.8. Find the Gradient of a Function

Question

Show that $\nabla f(r)=f^{\prime }(r)\frac{\mathbf{r}}{r}$, where $\mathbf{r}$ is the position vector $\mathbf{r}=(x_1,x_2,x_3)$ and $r=|\mathbf{r}|$.

Hint: recall that $f^{\prime }(r)=\frac{\partial f}{\partial r}$, then get to a step involving the expression $\frac{\partial r}{\partial x_i}$, and along the way write out an equation for $r$ in terms of $x_1,x_2,x_3$ (in suffix notation).

Use the definition of Gradient, then basic properties of position and partial derivatives, simplifying using Kronecker Delta to get the final answer.